I think the issue would be in the timing. You board a ship if you both touch it and defeat it. You touch it in the action phase and defeat it in the battle phase (or in the action phase as a separate action). When you defeat it and thereby board it, it is removed. Because other players also have the opportunity to touch it or defeat it in the action phase and potentially board it, it wouldn't do for you to be able to remove it immediately. I think the ship has to remain on the board until all the actions and battles are resolved.

Correct, all dice are resolved at the end. A good example of why this is needed, say you move and touch the NPC die to board it. Another player sees this and doesn't want you to get the points and loot, so they fire a cannon and sink the ship. Thus, they gain the coin and screw you in the process.

That all depends if they have cannons to fire or the ability to do so. Depending on how you build your ship, some players can have more shots to fire in a round than others. If someone shoots your ship and you have a shot you're able to take, you can definitely use your shot on another ship.

Shots are done in turn order as an action. If you move to touch an NPC ship, that's your action for the turn. If they fire on your ship, that's their action. When it's your turn, then you can use a cannon action, if you want to or are able to, on another NPC ship. Just because someone else uses their action doesn't give everyone else the opportunity to do the same immediately. Still need to follow turn order.